Just a thought but it might help if you approach this in another way and rather than getting bogged down with the CLB CON switch have a good look (if you can) at the autoflight takeoff profiles in the FCOM ( and Flight Crew Training Manual if you have it), especially the thrust modes. They explain in sequence what the system should do automatically and also in sequence what modes you'll get/see annunciated for both a normal take-off and an engine out. If you follow the profiles through you'll see CLB CON hardly (if ever) gets a mention...unless VNAV is inop or not selected - which is almost unheard of.
Otherwise it's as jammed stab describes..it's not now the newest aircraft on the block but it's got a fair degree of automation, even when "hand flying" the autothrottle is used and the system controls thrust/thrust selections in accordance with either what you've loaded in the FMS pre departure (e.g. Thrust reduction altitude) and/or various defaults, provided VNAV and Autothrottle are working normally. So on a departure with VNAV/autothrottle engaged and working if an engine fails the system "sees" the failure and changes it's thrust management to it's engine out schedule, giving you CON power automatically, with CON annunciated (at top of EICAS), once flaps are up and /or you are above the flaps maneuver speed....all without a button being pressed.
I'd also agree with
JS that generally the CLB CON button only really gets used if you want to intervene and set thrust at something other than non standard times/altitudes, which in turn leads to his point that:.
I suspect many pilots on type are not sure when it would be used(perhaps myself included) as it really never is used.
Agreed. It gets used in the sim on occasions, but I don't think I have ever used the CLB CON button on the line, that's why I'd recommend not getting bogged down in what it does ( and again - it does make much more sense in the sim).